Sigma-matrix ideals and Sigma-inverting homomorphisms

In Free Rings and their Relations, P.M.Cohn constructed a skew field from a prime matrix ideal using admissible matrices. When Σ is a lower multiplicative set of matrices over any non-commutative ring this method can be generalised to construct any epic Σ-inverting homomorphism upto isomorphism. This depends on the introduction of the concept of a Σ-matrix ideal. Every Σ-inverting homomorphism gives rise to a Σ-matrix ideal and conversely our main theorem shows that, given a Σ-matrix ideal, an epic Σ-inverting homomorphism can be constructed and that the matrices which are admissible for zero are precisely those lying in the Σ-matrix ideal. It is shown that the least Σ-matrix ideal induces the universal Σ-inverting homomorphism. A description of the least Σ-matrix ideal is then obtained yielding a new description of the kernel of the Σ-inverting homomorphism and a criterion for it to be an embedding; Malcolmson and Gerasimovs' respective descriptions of the kernel are also proved. When Σ is taken to be the complement of a prime matrix ideal the construction reduces to that used by Cohn to construct a skew field. It is further shown that the definition of a prime matrix ideal P can be simplified by restricting the class of matrices necessarily lying in P to be the hollow and degenerate matrices. The condition that P be closed with respect to row determinantal sums can be dropped completely. As a consequence, Cohn's criterion for the existence of a homomorphism from a ring to a field and the criterion for the existence of a field of fractions for a ring can be refined somewhat. For completeness Dicks and Sontag's result that Sylvester domains form the precise class of rings which have a universal field of fractions inverting all full matrices is also included